From 0e450644d85aa1fda15f2749b8cdf9f9d649e994 Mon Sep 17 00:00:00 2001 From: Nikola <2212719@stud.hs-mannheim.de> Date: Thu, 5 Nov 2026 19:18:10 +0100 Subject: [PATCH] add files for second testat --- Testat_2/Aufgabe_1/ZentraleGrenzwertsatz.Rmd | 50 ++++++ Testat_2/Aufgabe_2/tTestHeart.Rmd | 160 +++++++++++++++++++ Testat_2/Aufgabe_3/ULearningHeart.Rmd | 119 ++++++++++++++ Testat_2/Testat2.pdf | Bin 0 -> 346628 bytes 4 files changed, 329 insertions(+) create mode 100644 Testat_2/Aufgabe_1/ZentraleGrenzwertsatz.Rmd create mode 100644 Testat_2/Aufgabe_2/tTestHeart.Rmd create mode 100644 Testat_2/Aufgabe_3/ULearningHeart.Rmd create mode 100644 Testat_2/Testat2.pdf diff --git a/Testat_2/Aufgabe_1/ZentraleGrenzwertsatz.Rmd b/Testat_2/Aufgabe_1/ZentraleGrenzwertsatz.Rmd new file mode 100644 index 0000000..91db1ac --- /dev/null +++ b/Testat_2/Aufgabe_1/ZentraleGrenzwertsatz.Rmd @@ -0,0 +1,50 @@ +--- +title: "Zentrale Grenzwertsatz" +output: html_notebook +--- + +# Zentrale Grenzwertsatz + +Für jedes $n=1,2,\ldots$ seien die Zufallsvariablen $X_1, X_2, \ldots, X_n$ unabhängig und besitzen die gleiche Verteilung mit dem Erwartungswert $\mu = E(X_i)$ und der Varianz $\sigma^2=Var(X_i)$. Dann gilt für die Verteilungsfunktion der standardisierten Summen $G_n = \frac{\sum_{i=1}^{n}X_{i}-n\mu}{\sqrt{n}\sigma}$: +$$ +\lim_{n\rightarrow\infty}P\left(G_n \leq x\right)=\Phi(x) +$$ +Dabei ist $\Phi$ die Verteilungsfunktion der Standard-Normalverteilung $N(0,1)$. + +Überprüfen Sie mit R den zentralen Grenzwertsatz. + +## Verteilung + +Erzeugen Sie einen Data Frame, der die gezogenen Zufallszahlen $G_n$ enthält. Die Stichprobe soll einen Umfang von 50000 haben (Anzahl Zeilen der Tabelle). Der Data Frame hat sechs Spalten, die die Werte für die Zahlen $n=1, 2, 3, 10, 100, 1000$ enthalten. Geben Sie die ersten 10 Zeilen des Data Frames aus. + +Welche Verteilung Sie für $X_i$ nehmen, bleibt Ihnen überlassen. Es soll nur keine Normalverteilung sein. + + +```{r} +# Ihre Lösung: +``` + +## Plot + +Plotten Sie sechs Histogramme, die je für $n=1, 2, 3, 10, 100, 1000$ die Verteilung im Vergleich zu einer ($0,1$)-Normalverteilung zeigen. Die Intervallbreite soll 0,25 sein. + +Tipp: Platzieren Sie die sechs Plots auf einem $3 \times 2$-Gitter. + +```{r} +# Ihre Lösung: +``` + +## Abweichung + +Nun soll die Abweichung der Standardisierten $G$ von der Normalverteilung für alle $n$ quantifiziert werden. Hierzu soll für alle Balken (bins) der Histogramme aus der vorigen Aufgabe die Differenz von $G$ zur Normalverteilung gebildet und quadriert werden. Diese Werte werden aufaddiert und durch die Anzahl der Intervalle geteilt. Daraus wird die Wurzel gezogen. + +Sie können sich auch ein anderes Maß zur Bestimmung der Abweichung überlegen. + +Geben Sie die Abweichungen aus. Stimmt es, dass die Abweichungen mit größerem $n$ kleiner werden? + +Tipp: `hist(plot = FALSE)` erzeugt ein Histogramm, ohne es zu plotten. Gerne können Sie auch Ihr eigenes Histogramm nutzen. + +```{r} +# Ihre Lösung: +``` + diff --git a/Testat_2/Aufgabe_2/tTestHeart.Rmd b/Testat_2/Aufgabe_2/tTestHeart.Rmd new file mode 100644 index 0000000..dc5adc3 --- /dev/null +++ b/Testat_2/Aufgabe_2/tTestHeart.Rmd @@ -0,0 +1,160 @@ +--- +title: "Untersuchungen zur koronaren Herzkrankheit (t-tests)" +output: html_document +--- + +```{r echo = F, warning = F, message = F} +library(knitr) +# Laden Sie benötigte Libraries... +``` +# Untersuchungen zur koronaren Herzkrankheit + +In diesem Abschnitt sollen Daten von Probanden bzw. Patienten auf das Risiko für koronare Herzkrankheit untersucht werden. Dies ist eine Erkrankung der Herzkranzgefäße (Koronararterien), die sich durch Ablagerungen in den Gefäßwänden verengen. Der Original-Herz-Datensatz ist unter + + * https://archive.ics.uci.edu/ml/datasets/Heart+Disease + +beschrieben. Wir nutzen eine konsolidierte CSV-Datei, die bereits Header enthält. Download unter: + + * https://oc.informatik.hs-mannheim.de/s/wyzFq34K9HiNjXR/download + +Die Datei enthält 13 Merkmale, die einen Einfluss auf eine koronare Herzkrankheit haben können. Das 14. Merkmal `goal` (im Original auch `num`) ist die Diagnose (Klassifizierung). Der Wert ist 0, falls keine krankhafte Verengung der Gefäße vorliegt, oder 1, 2, 3 oder 4, falls -- je nach Stärke -- eine krankhafte Verengung der Gefäße vorliegt. Wir unterscheiden im Folgenden nur die Zustände "gesund" (0) und "krank" (1, 2, 3, 4). Unter + + * https://archive.ics.uci.edu/ml/machine-learning-databases/heart-disease/heart-disease.names + +finden Sie eine Beschreibung aller Attribute. Hier ist eine Zusammenfassung. Wir benötigen insbesondere die Merkmale `sex`, `trestbps`, `chol` und `goal`. + +|Feld|Bedeutung| +|-|------| +| age | age in years | +| sex | sex (1 = male; 0 = female)| +| cp | chest pain type (1 = typical angina; 2 = atypical angina; 3 = non-anginal pain; 4 = asymptomatic) | +| trestbps | resting systolic blood pressure (in mmHg on admission to the hospital) | +| chol | serum cholestoral in mg/dl | +| fbs | fasting blood sugar > 120 mg/dl (1 = true; 0 = false) | +| restecg | resting electrocardiographic results (0 = normal; 1 = having ST-T wave abnormality (T wave inversions and/or ST elevation or depression of > 0.05 mV); 2 = showing probable or definite left ventricular hypertrophy by Estes' criteria) | +| thalach | maximum heart rate achieved | +| exang | exercise induced angina (1 = yes; 0 = no) | +| oldpeak | ST depression induced by exercise relative to rest.^[ST depression refers to a finding on an electrocardiogram, wherein the trace in the ST segment is abnormally low below the baseline.] | +| slope | slope of the peak exercise ST segment (1 = upsloping; 2 = flat; 3 = downsloping) | +| ca | number of major vessels (0-3) colored by flourosopy | +| thal | 3 = normal; 6 = fixed defect; 7 = reversable defect | +| goal | diagnosis of heart disease (0: < 50% diameter narrowing ; 1, 2, 3, 4: > 50% diameter narrowing) | + +## Einlesen der Herz-Daten + +Lesen Sie die Datei aus der URL als Data Frame zur weiteren Bearbeitung ein. Überlegen Sie, ob sie Faktoren sinnvoll einsetzen können. Geben Sie die ersten drei Zeilen und fünf Spalten aus^[Möglicherweise kommt es zu einem Fehler beim Einlesen des ersten Attributs (`age`). Manuelles Umbenennen hilft.]: + +```{r} +# Ihre Lösung: +``` + +## Cholesterin im Vergleich Männer/Frauen + +Nun sollen die Cholerstin-Werte untersucht werden -- zunächst im Vergleich Männer zu Frauen. + +### Überblick über Cholesterin-Daten + +Verschaffen Sie sich einen Überblick, indem Sie ein Boxplot für das Cholesterin gruppiert nach dem Geschlecht plotten. + +```{r} +# Ihre Lösung: +``` + +### Konfidenz-Intervall + +Berechnen Sie das Konfidenz-Intervall (Niveau 95%) für den Cholesterin-Level jeweils für Männer und Frauen. + +#### Tabelle + +Geben Sie das Ergebnis als `kable`-Tabelle aus: + +```{r} +# Ihre Lösung: +``` + +#### Überlappung? + +Überlappen sich die Bereiche? + +## Cholesterin im Vergleich zur Erkrankung + +Nun sollen die Cholesterin-Werte in Abhängigkeit der Diagnose untersucht werden. + +### Überblick über Cholesterin-Daten + +Verschaffen Sie sich einen Überblick, indem Sie ein Boxplot für das Cholesterin gruppiert nach der Diagnose plotten. + +```{r} +# Ihre Lösung: +``` + +### Konfidenz-Intervall + +Berechnen Sie die Konfidenzintervalle für beide Gruppen und geben Sie das Ergebnis als `kable`-Tabelle aus: + +```{r} +# Ihre Lösung: +``` + +### Test + +Es sieht so aus, als ob der Cholesterin-Wert bei den erkrankten Patienten höher ist als bei den nicht erkrankten. Überprüfen Sie das mit einem Hypothesen-Test. + +#### Wie lauten die Hypothesen? + +Formulieren Sie die Hypothesen ($H_0$ und $H_1$). + + +#### Testanwendung + +Wenden Sie den Test mit R an. Was ist das Ergebnis? + +```{r} +# Ihre Lösung: +``` + + + +## Systolischer Ruheblutdruck + +Der systolische Blutdruck liegt beim gesunden Menschen bei ca. 120 mmHg. + +### Überblick über Blutdruck + +#### Plot + +Verschaffen Sie sich einen Überblick, indem Sie ein Boxplot für den Blutdruck in Ruhe gruppiert nach der Diagnose plotten. + +```{r} +# Ihre Lösung: +``` + +#### Normalverteilt? + +Kann überhaupt davon ausgegangen werden, dass die Daten normalverteilt sind? + +### Konfidenzintervalle nach Erkrankung + +Berechnen Sie die Konfidenzintervalle für den Ruheblutdruck aufgeschlüsselt nach der Diagnose (erkrankt/nicht erkrankt) und geben Sie das Ergebnis als `kable`-Tabelle aus: + +```{r} +# Ihre Lösung: +``` + +### Test, ob Kranke höheren Ruhe-Blutdruck haben + +Überprüfen Sie mit einem Hypothesen-Test, ob Erkrankte einen höheren Ruhe-Blutdruck haben als gesunde Probanden. + +#### Wie lauten die Hypothesen? + +Formulieren Sie die Hypothesen ($H_0$ und $H_1$). + + +#### Testanwendung + +Wenden Sie den Test mit R an. Was ist das Ergebnis? + +```{r} +# Ihre Lösung: +``` + diff --git a/Testat_2/Aufgabe_3/ULearningHeart.Rmd b/Testat_2/Aufgabe_3/ULearningHeart.Rmd new file mode 100644 index 0000000..01408a3 --- /dev/null +++ b/Testat_2/Aufgabe_3/ULearningHeart.Rmd @@ -0,0 +1,119 @@ +--- +title: "Unüberwachtes Lernen mit dem Herz-Datensatz" +output: html_document +--- + +```{r echo = F, warning = F, message = F} +# Laden Sie benötigte Libraries... +``` + +# Clustering und PCA auf die Herzdaten + +## Einlesen der Herz-Daten + +Es werden wieder die Herzdaten aus der letzten Aufgabe genutzt. Lesen Sie diese als Data Frame ein. + +```{r} +# Ihre Lösung: +``` + +## Bedeutet "ähnliche Merkmale" auch "gleiche Diagnose"? + +Für jeden Datensatz ist bekannt, zu welcher Klasse er gehört: 0 (gesund) und 1 (erkrankt). Wir wollen untersuchen, wie gut _ähnliche_ Datensätze zur gleichen Klasse gehören. Dafür soll mit dem $k$-means-Clusterverfahren der Datensatz in zwei Cluster eingeteilt werden. + +### Nur reelle Merkmale + +Zunächst sollen **nur die nummerischen Merkmale** benutzt werden und nicht jene, die Faktoren sind. + +#### Clustering + +Clustern Sie diese Daten. Überlegen Sie, ob Sie die Daten standardisieren wollen. + +```{r} +# Ihre Lösung: +``` + +#### Richtig? + +Berechnen Sie, wie viel Prozent der Datensätze richtig einem Cluster eingeordnet wurden und geben Sie die Zahl auf zwei Nachkommastellen gerundet aus. + +Hinweis: Berücksichtigen Sie, dass die Vergabe der Clusternummern zufällig ist. D.h. sowohl die Cluster (1, 2) wie auch (2, 1) sind möglich. + +```{r} +# Ihre Lösung: +``` + +#### Scatterplot age vs. thalach + +Plotten Sie die Merkmale `age` und `thalach` als Scatterplot. Färben Sie die Punkte gemäß ihrer Clusterzuordnung ein. Die Form (`shape`) eines Punkts soll zeigen, ob die Klassifikation (d.h. der Cluster) richtig oder falsch ist. + +```{r} +# Ihre Lösung: +``` + + +### Mit Dummy-Variablen + +Nun sollen **alle Merkmale** benutzt werden. + +#### Clustering + +Clustern Sie diese Daten. Überlegen Sie, wie die Faktoren zu Zahlen werden. + +```{r} +# Ihre Lösung: +``` + +#### Richtig? + +Berechnen Sie für diesen Fall, wie viel Prozent der Datensätze richtig einem Cluster eingeordnet wurden und geben Sie die Zahl auf zwei Nachkommastellen gerundet aus. Wie hat sich der Wert verändert? Warum ist dies so? + +```{r} +# Ihre Lösung: +``` + +#### Scatterplot age vs. thalach + +Plotten Sie erneut und schauen Sie, wie die richtigen nun Punkte verteilt sind. + +```{r} +# Ihre Lösung: +``` + +## PCA + +Wenden Sie eine PCA auf diesen Datensatz an. Es sollen alle Merkmale berücksichtigt werden. + +### Wichtige Merkmale + +Welche Merkmale der ersten Hauptkomponente tragen am meisten zur Varianz bei? Geben Sie die TOP-10-Merkmale an. + +```{r} +# Ihre Lösung: +``` + +### Erste und zweite Hauptkomponente + +Plotten Sie die erste und zweite Hauptkomponente als Scatterplot. Färben Sie die Punkte gemäß ihrer Klasse (Disease) ein. + +```{r} +# Ihre Lösung: +``` + +### PVE + +#### Plot + +Plotten Sie die Proportion of Variance explained (PVE) für jede Hauptkomponente sowie die akkumulierte PVE. + +```{r echo=F} +# Ihre Lösung +``` + +#### Wichtige Hauptkomponenten + +Wie viele Hauptkomponenten erklären mehr als 50% der Varianz? + + +Möglicherweise tragen bei Ihrem Ergebnis die letzten Hauptkomponenten keine Varianz mehr bei. 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